Micale et al. Appl Netw Sci (2021) 6:55 https://doi.org/10.1007/s41109-021-00397-0 Applied Network Science RESEARCH Open Access TemporalRI: subgraph isomorphism in temporal networks with multiple contacts Giovanni Micale1* , Giorgio Locicero2, Alfredo Pulvirenti1 and Alfredo Ferro1 *Correspondence: [email protected] Abstract 1 Department of Clinical Temporal networks are graphs where each edge is associated with a timestamp denot- and Experimental Medicine, University of Catania, Catania, ing when two nodes interact. Temporal Subgraph Isomorphism (TSI) aims at retrieving Italy all the subgraphs of a temporal network (called target) matching a smaller temporal Full list of author information network (called query), such that matched target edges appear in the same chrono- is available at the end of the article logical order of corresponding query edges. Few algorithms have been proposed to solve the TSI problem (or variants of it) and most of them are applicable only to small or specifc queries. In this paper we present TemporalRI, a new subgraph isomorphism algorithm for temporal networks with multiple contacts between nodes, which is inspired by RI algorithm. TemporalRI introduces the notion of temporal fows and uses them to flter the search space of candidate nodes for the matching. Our algorithm can handle queries of any size and any topology. Experiments on real networks of diferent sizes show that TemporalRI is very efcient compared to the state-of-the-art, especially for large queries and targets. Keywords: Subgraph isomorphism, Temporal networks, Dynamic networks, Network analysis Introduction and related works Graphs (or networks) are mathematical objects that are suitable to represent complex systems formed by a set of entities that interact each other. Entities are called nodes while their interactions are called edges. In many applications graphs are considered as static objects, without taking into account when or how long two nodes interact. How- ever, complex systems are inherently dynamic and evolve during time. For example, in a remote communication system, users may enter the network anytime to start commu- nications with other users. In a protein-protein interaction network a protein can estab- lish temporary interactions with one or more proteins to perform a biological process or transmit a signal to a cell. Terefore, time is crucial to understand the formation and the evolution of such systems. By associating a time information to each edge, a network becomes temporal. Tere are several defnitions of temporal networks in literature (Holme and Saramaki 2012; Masuda and Lambiotte 2020), which are commonly referred to as dynamic (Car- ley et al. 2007), evolutionary (Aggarwal and Subbian 2014) or time-varying (Casteigts et al. 2011). In this paper, we defne temporal network as a multigraph (i.e a graph with © The Author(s), 2021. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the mate- rial. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Micale et al. Appl Netw Sci (2021) 6:55 Page 2 of 22 one or more edges between any two nodes) where each edge is associated with an inte- ger, called timestamp, denoting when two nodes interact. Several frameworks have been proposed to study properties of temporal networks. Analysis of temporal networks includes network centrality (Lv et al. 2019; Tsalouchidou et al. 2020), network clustering (Crawford and Milenkovic 2018), community detection (Rossetti and Cazabet 2018), link prediction (Divakaran and Mohan 2020), graph min- ing (Sun et al. 2019a), graph embedding (Torricelli et al. 2020), network sampling (Rocha et al. 2017), random models (Singh and Cherif 2020; Petit et al. 2018; Hiraoka et al. 2020) and epidemic spreading (Tizzani et al. 2018; Masuda and Holme 2020; Williams et al. 2019). Comprehensive reviews of temporal networks and their main properties can be found in Holme and Saramaki (2012, 2019) and Masuda and Lambiotte (2020). Here, we focus on Temporal Subgraph Isomorphism (TSI) problem. Given two tem- poral graphs Q and T, called query and target, respectively, and a time interval , TSI problem aims at fnding a subgraph S of T (called occurrence of Q in T) such that: (1) Q and S are isomorphic, i.e. structurally equivalent, (2) edges in S follow the same chrono- logical order imposed by corresponding matched edges in Q, (3) all interactions in T are observed in a time interval less than or equal to . Te problem can have more than one solution, i.e. there can exist two or more occurrences of Q in T. Figure 1 shows a toy example of TSI. Subgraph isomorphism problem in static graphs have been widely investigated and several methods have been presented (Cordella et al. 2004; Carletti et al. 2017; Bonnici et al. 2013; Bonnici and Giugno 2017; Han et al. 2019; Sun and Luo 2020; Han et al. 2013; Bi et al. 2016). However, as far as we know, few algorithms have been proposed for TSI or similar defnitions of the problem (Redmond and Cunningham 2013b, 2016; Mackey et al. 2018; Sun et al. 2019b; Kim et al. 2018). Redmond and Cunningham (2013b, 2016) introduce for the frst time the TSI prob- lem, where: (1) queries are static graphs with no timestamps, (2) all paths in the matched target subgraph must be time-respecting, i.e. any outcoming event from a node has to follow any incoming event from the same node, and (3) consecutive events are d-adja- cent, i.e. the diference between their timestamps must not exceed a threshold d. Te authors also presented a modifed version of the subgraph isomorphism algorithm VF2 (Cordella et al. 2004) capable to check the time constraints of the query during the search. Te same defnition of the TSI problem we introduced above is given in Mackey et al. (2018) and Sun et al. (2019b). Mackey et al. (2018) describe a general algorithm which orders both query and target edges before matching. By doing so, whenever a new match between a target edge t and a query edge q is found, the algorithm can continue the search from the target edge following t in the ordering. Sun et al. (2019b) illustrate a method for counting the number of occurrences of a temporal query in a temporal target, based on partitioning the graph into subgraphs and then counting the pattern in each subgraph while matching motif topology. A problem closely related to TSI is temporal motifs search, which consists in fnding all small and recurrent temporal subgraphs of interactions, called motifs, having k nodes and/or m edges (Kovanen et al. 2011; Paranjape et al. 2017; Liu et al. 2019; Hulovatyy et al. 2015). Te main diference between TSI and motif search is that TSI enumerates all the subgraphs matching a specifc query, while motif search counts the occurrences Micale et al. Appl Netw Sci (2021) 6:55 Page 3 of 22 of all possible subgraphs with a specifed number of nodes and edges. Due to the compu- tational complexity of the problem, motif search algorithms usually focus on very small motifs or on specifc topologies. Temporal motifs were introduced for the frst time by Kovanen et al. (2011) and are defned as ordered sets of events such that: (1) the time dif- ference between two consecutive events is within a threshold t and (2) adjacent events, i.e. events sharing a node, are consecutive, so the node cannot participate in any other event in the meanwhile. Hulovatyy et al. (2015) relax the latter constraint and introduce the concept of dynamic graphlets to capture how the neighborhood of a node changes over time. Tis is done to reduce the computational complexity while obtaining approxi- mate results. As in Mackey et al. (2018) and Sun et al. (2019b), Paranjape et al. (2017) defne a temporal motif as a subgraph with a temporal order of edges. Te authors pre- sent an algorithm to efciently calculate the frequencies of 2-node, 2-stars and 3-node triangle temporal motifs. For bigger motifs they use a naive algorithm that frst com- putes static matches, then flters out occurrences which do not match the temporal constraints. To tackle with the NP-completeness of temporal motif search problem, Liu et al. (2019) propose a general sampling framework to estimate motif counts. It consists in partitioning time into intervals, fnding exact counts of motifs in each interval and weighting counts to get the fnal estimate, using importance sampling. Here, we present a new algorithm for the TSI problem, called TemporalRI, inspired by RI algorithm for subgraph matching in static networks (Bonnici et al. 2013; Bonnici and Giugno 2017). TemporalRI can be applied to queries of any size, in terms of num- ber of nodes and edges. Tis paper extends in several directions the preliminary work presented in Locicero et al. (2021) where a frst implementation of TemporalRI was presented. From a theoretical point of view, the defnition of TSI has been refned by introducing a parameter to set a maximum temporal interval in which interactions should be observed.